Properties

Label 966.17
Modulus 966966
Conductor 483483
Order 6666
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(966, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([33,11,21]))
 
Copy content pari:[g,chi] = znchar(Mod(17,966))
 

Basic properties

Modulus: 966966
Conductor: 483483
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 6666
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from χ483(17,)\chi_{483}(17,\cdot)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 966.bc

χ966(5,)\chi_{966}(5,\cdot) χ966(17,)\chi_{966}(17,\cdot) χ966(89,)\chi_{966}(89,\cdot) χ966(143,)\chi_{966}(143,\cdot) χ966(227,)\chi_{966}(227,\cdot) χ966(341,)\chi_{966}(341,\cdot) χ966(383,)\chi_{966}(383,\cdot) χ966(425,)\chi_{966}(425,\cdot) χ966(467,)\chi_{966}(467,\cdot) χ966(479,)\chi_{966}(479,\cdot) χ966(521,)\chi_{966}(521,\cdot) χ966(563,)\chi_{966}(563,\cdot) χ966(605,)\chi_{966}(605,\cdot) χ966(635,)\chi_{966}(635,\cdot) χ966(677,)\chi_{966}(677,\cdot) χ966(773,)\chi_{966}(773,\cdot) χ966(803,)\chi_{966}(803,\cdot) χ966(815,)\chi_{966}(815,\cdot) χ966(845,)\chi_{966}(845,\cdot) χ966(941,)\chi_{966}(941,\cdot)

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: Q(ζ33)\Q(\zeta_{33})
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

(323,829,925)(323,829,925)(1,e(16),e(722))(-1,e\left(\frac{1}{6}\right),e\left(\frac{7}{22}\right))

First values

aa 1-11155111113131717191925252929313137374141
χ966(17,a) \chi_{ 966 }(17, a) 1-111e(4366)e\left(\frac{43}{66}\right)e(133)e\left(\frac{1}{33}\right)e(2122)e\left(\frac{21}{22}\right)e(5966)e\left(\frac{59}{66}\right)e(2033)e\left(\frac{20}{33}\right)e(1033)e\left(\frac{10}{33}\right)e(522)e\left(\frac{5}{22}\right)e(566)e\left(\frac{5}{66}\right)e(166)e\left(\frac{1}{66}\right)e(911)e\left(\frac{9}{11}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ966(17,a)   \chi_{ 966 }(17,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ966(17,))   \tau_{ a }( \chi_{ 966 }(17,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ966(17,),χ966(n,))   J(\chi_{ 966 }(17,·),\chi_{ 966 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ966(17,))  K(a,b,\chi_{ 966 }(17,·)) \; at   a,b=\; a,b = e.g. 1,2